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Theorem ordtri3 4428
Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )

Proof of Theorem ordtri3
StepHypRef Expression
1 ordirr 4410 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq2 2344 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
32notbid 285 . . . . . 6  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  A  e.  B ) )
41, 3syl5ib 210 . . . . 5  |-  ( A  =  B  ->  ( Ord  A  ->  -.  A  e.  B ) )
5 ordirr 4410 . . . . . 6  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2344 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 285 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibr 212 . . . . 5  |-  ( A  =  B  ->  ( Ord  B  ->  -.  B  e.  A ) )
94, 8anim12d 546 . . . 4  |-  ( A  =  B  ->  (
( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
109com12 27 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
11 pm4.56 481 . . 3  |-  ( ( -.  A  e.  B  /\  -.  B  e.  A
)  <->  -.  ( A  e.  B  \/  B  e.  A ) )
1210, 11syl6ib 217 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  -.  ( A  e.  B  \/  B  e.  A )
) )
13 ordtri3or 4424 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
14 df-3or 935 . . . . 5  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1513, 14sylib 188 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
16 or32 513 . . . 4  |-  ( ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A )  <->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1715, 16sylib 188 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1817ord 366 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  e.  B  \/  B  e.  A
)  ->  A  =  B ) )
1912, 18impbid 183 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   Ord word 4391
This theorem is referenced by:  ordunisuc2  4635  tz7.48lem  6453  oacan  6546  omcan  6567  oecan  6587  omsmo  6652  omopthi  6655  inf3lem6  7334  cantnfp1lem3  7382  infpssrlem5  7933  fin23lem24  7948  isf32lem4  7982  om2uzf1oi  11016  nodenselem4  24338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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